The main difference is sorting accuracy (when comparing characters in the language) and performance. The only special one is utf8_bin which is for comparing characters in binary format.

utf8_general_ci is somewhat faster than utf8_unicode_ci, but less accurate (for sorting). The specific language utf8 encoding (such as utf8_swedish_ci) contain additional language rules that make them the most accurate to sort for those languages. Most of the time I use utf8_unicode_ci (I prefer accuracy to small performance improvements), unless I have a good reason to prefer a specific language.

Primary key is usually used to create a numerical 'id' for your records, and this id column is automatically incremented.

For example, if you have a books table with an id field, where the id is the primary key and is also set to auto_increment (Under 'Extra in phpmyadmin), then when you first add a book to the table, the id for that will become 1'. The next book's id would automatically be '2', and so on. Normally, every table should have at least one primary key to help identifying and finding records easily.

Indexes are used when you need to retrieve certain information from a table regularly. For example, if you have a users table, and you will need to access the email column a lot, then you can add an index on email, and this will cause queries accessing the email to be faster.

However there are also downsides for adding unnecessary indexes, so add this only on the columns that really do need to be accessed more than the others. For example, UPDATE, DELETE and INSERT queries will be a little slower the more indexes you have, as MySQL needs to store extra information for each indexed column. More info can be found at this page.

Edit: Yes, columns that need to be used in ORDER BY a lot should have indexes, as well as those used in WHERE.